Global dissipativity and inertial manifolds for diffusive burgers equations with low-wavenumber instability

Abstract

Global well-posedness, existence of globally absorbing sets and existence of inertial manifolds is investigated for a class of diffusive Burgers equations. The class includes diffusive Burgers equation with nontrivial forcing, the Burgers-Sivashinsky equation and the Quasi-Stedy equation of cellular flames. The global dissipativity is proven in 2D for periodic boundary conditions. For the proof of the existence of inertial manifolds, the spectral-gap condition, which Burgers-type equations do not satisfy in its original form is circumvented by the Cole-Hopf transform. The procedure is valid in both one and two space dimensions.

Kuramoto-Sivashinsky Equation and Free-Interface Models in Combustion Theory

“…hence, the convergence to K-S, see [2,Section 3]. From a more physical viewpoint, all three of these models, K-S, κ−θ and Q-S share the same basic quality revealed by linear stability analysis, namely long-wave destabilization, which is suppressed by the dominant dissipative principal term for small wave lengths (see the discussion in [3] and Vukadinovic [36]).…”

“…Although the Q-S equation was introduced as an ad hoc truncation of the full κ−θ model (2.2), it represents an interesting dynamical system in its own right. Its dynamics is essentially finite-dimensional: Q-S possesses a universal absorbing set and a compact attractor; furthermore, the attractor is of a finite Hausdorff dimension (see [4] and Vukadinovic [36]).…”

A parabolic equation with nonlocal diffusion without a smooth inertial manifold

“…If the linear part −A of the parabolic equation (1.1) is the Laplace operator ∆ with standard boundary conditions in L 2 (Ω), Ω ⊆ R m , then these conditions become restrictive owing to the well-known asymptotics λ n ∼ cn 2/m of the eigenvalues λ n ∈ σ(−∆). Attempts to sidestep condition (2.2) have only been successful in isolated special cases (e.g., see [10,11]). By now, the asymptotic finite-dimensionality has not been established even for relatively simple problems such as the parabolic equation…”

Параболическое Уравнение С Нелокальной Диффузией Без Гладкого Инерциального Многообразия

“…Попытки обойти условие (2.2) приводят к успеху (см., например, [10], [11]) лишь в отдельных специальных случаях. К настоящему времени асимптотическая конечномерность не установлена даже для столь простых задач как параболическое уравнение вида…”